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arXiv · 1404.7417

Bifurcation measures and quadratic rational maps

Laura DeMarco·Xiaoguang Wang·Hexi Ye
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Abstract

We study critical orbits and bifurcations within the moduli space of quadratic rational maps on $\mathbb{P}^1$. We focus on the family of curves, $Per_1(λ)$ for $λ$ in $\mathbb{C}$, defined by the condition that each $f\in Per_1(λ)$ has a fixed point of multiplier $λ$. We prove that the curve $Per_1(λ)$ contains infinitely many postcritically-finite maps if and only if $λ= 0$; addressing a special case of [BD2, Conjecture 1.4]. We also show that the two critical points of a map $f$ define distinct bifurcation measures along $Per_1(λ)$.

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Laura DeMarco, Xiaoguang Wang, Hexi Ye. 2015-06-16. Bifurcation measures and quadratic rational maps. https://doi.org/10.1112/plms%2Fpdv024

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